By solving the maximization problem, characterize the saving function depending on the value of 𝜃, i.e., there are three cases.

Macroeconomics
Tomohiro Hirano
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Problem 1
We consider a CES production function.
𝑌𝑡 = 𝐴 (𝛼𝐾𝑡
𝜎−1
𝜎 + (1 − 𝛼)𝐿𝑡
𝜎−1
𝜎 )
𝜎
𝜎−1
.
Q1: As 𝜎 → 1, prove the Cobb-Douglas production function 𝐴𝐾𝑡
𝛼𝐿𝑡
1−𝛼. (10 marks)
Q2: As 𝜎 → 0, prove the Leontief production function 𝑌𝑡 = 𝐴 min(𝐾𝑡, 𝐿𝑡). (10 marks)
Q3: The profit maximization problem is given by
max
𝐾𝑡,𝐿𝑡
𝜋𝑡 = 𝑌𝑡 − 𝑅𝑡𝐾𝑡 − 𝑤𝑡𝐿𝑡
By solving the profit maximization problem, derive the definition of the value of 𝜎
mathematically. (10 marks)
Problem 2
The utility maximization problem is given by
max
𝑐1𝑡,𝑐2𝑡,𝑠𝑡
𝑢𝑡 = (𝑎1
1
𝜃(𝑐1𝑡)𝜃−1
𝜃 + 𝑎2
1
𝜃(𝑐2𝑡)𝜃−1
𝜃 )
𝜃
𝜃−1
subject to
𝑐1𝑡 + 𝑠𝑡 = 𝑤𝑡 + 𝑒
𝑐2𝑡 = (1 + 𝑟𝑡+1)𝑠𝑡
Q4: By solving the maximization problem, characterize the saving function depending on the
value of 𝜃, i.e., there are three cases. (30 marks)
Q5: By solving the maximization problem, derive the definition of the value of 𝜃
mathematically. (10 marks)
Problem 3
Consider a CES utility function.
𝑢𝑡 = (𝑎1
1
𝜃(𝑐1𝑡)𝜃−1
𝜃 + 𝑎2
1
𝜃(𝑐2𝑡)𝜃−1
𝜃 )
𝜃
𝜃−1
Q6: Derive 𝑢𝑡 as 𝜃 → 1. (10 marks)
Problem 4
Consider the following CES production function.
𝑌𝑡 = 𝐴 (𝛼 (𝐾𝑡
ℎ1
)
𝜎−1
𝜎
+ (1 − 𝛼) (𝐿𝑡
ℎ2
)
𝜎−1
𝜎
)
𝜎
𝜎−1
Q7: Derive factor prices 𝑅𝑡 and 𝑤𝑡. (10 marks)
Q8: Compute the values of 𝑅𝑡 and 𝑤𝑡, respectively, as 𝜎 → 0. (10 mark